Question: Simplify the following expression and state the condition under which the simplification is valid. $t = \dfrac{p^2 - 64}{p + 8}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = p$ $ b = \sqrt{64} = 8$ So we can rewrite the expression as: $t = \dfrac{({p} + {8})({p} {-8})} {p + 8} $ We can divide the numerator and denominator by $(p + 8)$ on condition that $p \neq -8$ Therefore $t = p - 8; p \neq -8$